hendric, the raw count of observed candidates is biased in two (and a half) ways from the actual frequency of planets in a given bin:

1) The geometric bias takes into account that for most orbital inclinations, transits will never be observable. The extent of this is a factor of distance from the primary. Happily, we can model this exactly: It's straight-up Euclidean geometry. A planet orbiting at 1AU has a probability of 0.29% of transiting a sun-sized star. For each distance we consider, this bias can be calculated exactly. If we observe N planets at 1AU, and that count is statistically significant, we can be sure that about (N / 0.29%) planets exist (and we missed almost all of them).

It may help to imagine if, say, Mercury and Neptune were in the exact same plane, and for some observer, Mercury transited the Sun but appeared to slice across latitude 60 North. For that observer, Neptune will fail to transit by a great margin, several Sun radii, passing far above the Sun's disc each time it passed across the Sun's central meridian.

1 and a half) Planets in further-out orbits orbit more slowly, so the time required to witness N transits will vary with the orbital period. This, too, is pretty easy to model. It also, in many cases, does not matter, if the ratio of the observation period to the orbital period is high. For periods of, say, two days, the planet has had hundreds of chances to transit: If it's going to happen, it's already happened a lot. For periods of, say, ten years, this factor is quite significant, because while the team could report a giant planet very far out on the basis of a single transit, most such planets have not had a chance to perform their transit. 15 of the published candidates actually have orbital periods longer than the observation period: For each such observation, there are likely (far) more at that distance that will happen but haven't had a chance to yet.

2) Noise bias: Variability in the star and our measurements exist: When a large planet transits, it blots out a larger area with almost perfect blackness than when a smaller planet transits. With smaller planets, the transit is insignificant compared to the star's baseline variability, and in such cases, detection is impossible. The devil is in the details here, and the noise happens to be greater, for many stars, than we expected.

For each size bin, I calculated this by taking the SNR of observed large planets. Each such positive event allows us to say how much smaller of a planet *could* have been detected, with its correspondingly smaller transit effect. By sampling all of the largest planet candidates, I got a distribution of star noise. For each candidate size, we can say what fraction of stars allow detection of that size. This is another de-bias factor, a bit less certain than the geometric one, but it should be correct to a first order. I also performed this with the four-month release and found very similar values. We can't be totally sure that the stars hosting larger planets are precisely the same as the stars which do not have larger planets... that is an unconstrained possibility, but there's no reason, either, to assume that they are different than the other stars with respect to noise.

For each bin, then, we have an expected bias factor: What fraction of existing planets we would expect to find. By multiplying the observed candidates by the reciprocal of the bias factor, we get an estimate of the total number of planets in that bin, counting all of the ones we missed because of misalignment or noise.

3/ I've heard it stated (in the infamous IAU debate on the definition of "planet") that the solar system is dynamically 'full' in the sense that the planets are packed together as closely as possible apart from the gap where the asteroid belt is located. The dense compact systems Kepler has found seem to contradict this hypothesis, unless in fact they are young systems that will in due course eject or swallow most of the objects now observed.

There's a gaping hole inward of Mercury that is oddly void of planets. Kepler and HARPS have shown us that low-mass planets are common in this region, so our solar system seems slightly odd in this respect, too, regarding the "packed planetary system hypothesis." It's also worth noting that the Kepler planets, being typically closer to the star than the planets in our solar system, will have smaller Hill Spheres, and can therefore be packed in tighter than what you see for the Solar System without destabilising.

You can also look at the satellite systems of Jupiter, Saturn, and Uranus: Packed in very tight. It's not absolute worlds per orbital-radius that poses a limit, but a more complex relationship between the distances and masses.

The relationship between neighboring planets as seen in Kepler data is also interesting. The ratio of orbital periods peaks sharply at 1.5, with a resonance that is seen in our solar system between Neptune and KBOs, but not among the inner/major planets. There is also a desert at a ratio of about 1.97 picking up against at/after 2.0.

The kicker in your analysis is the assumption that planetary systems are a single normally distributed population. As you say in your piece a second distinct population with loads of earth-analogues could be in there, we simply don't know at this point.

My understanding of current planetary systems formation theories is that the predominant planet (e.g. Jupiter) migrates inward and pushes all other planets closer to the star; hence all the "hot Jupiters". In case of the solar system, Saturn came to the rescue and the gravitational interaction between Saturn and Jupiter stopped the inward migration early and there might even caused a phase of outward migration. I don't know if there is any evidence to support this theory in Kepler data.

When the big planet forms (the so called oligarchs), it cleans a large swath of the protoplanetary disk and slows the growth of any other planet in the vicinity. And when it migrates inward, the "vicinity" is a very large area.

Blair, there are unfortunately a lot of kickers... Wes Traub's paper lists a lot of possible factors which are currently unconstrained and which he assumes, therefore, to be nonexistent. I happened to include two factors that he did not:

F3 Class of star(But note, the majority of stars in the release are fairly sunlike.)

And there are many more factors which are unconstrained in either study. Particularly, those regarding the variation in noise across various circumstances are potentially very thorny. We might hope and expect that that factor is not psychopathic with respect to any key independent variables, but it's a risky assumption. Any extrapolation is: There could be a second peak in the distribution for any of these planet sizes: We also have yet to prove that the number of Earth-sized planets orbiting at Earth-like distances from Sun-like stars is not arbitrarily close to zero! (In fact, Venus is still the best Earth surrogate we've actually found.)

I will take some effort to model the known sources of variation in more detail. Two that I think are particularly worthwhile are F3 and

F4 The effect of transit "latitude" on SNR: Transits which skim the top/bottom of the stellar disc feature less signal and perhaps different distributions of noise.

Conceivably, we may see important cumulative effects in the bias that either raise or lower the estimation of absolute frequencies. I think the main qualitative result of my analysis will stand: Terrestrial planet frequency drops off with distance much faster than giant planet frequency; but the quantitative models may merit considerable revision. I'll be working on this, and I'm sure I won't be alone.

"F4 The effect of transit "latitude" on SNR: Transits which skim the top/bottom of the stellar disc feature less signal and perhaps different distributions of noise."

I've been curious how transit latitude can be ascertained. I keep googling my questions, always get pointed to this thread, lol

Question: Might not a tiny planet making a quick equatorial transit create a similar transit signal to a larger planet in a longer orbit skimming the top/bottom of the stellar disc? Would subsequent observations from land-based scopes detect a larger wobble from the latter?

Nice thought: Similar to the Hubble database, as time goes on and software improves, the database established by Kepler may subsequently reveal more planets, especially the harder-to-find nuggets we're hoping to identify.

Might not a tiny planet making a quick equatorial transit create a similar transit signal to a larger planet in a longer orbit skimming the top/bottom of the stellar disc?

The same transit depth, but the shape of the transit curve is different. For an equitorial transit, the transit curve is "flat bottom", whereas a skimming transit curve looks "round". Of course, you need a good signal to noise ratio to get the shape of the transit curve. On the other hand, if SNR is really good, one may also model smaller effects like limb darkening and such.

Not to mention that if we have the orbital period, we would know how long an equatorial transit "should" take, assuming the stellar diameter expected for the star's brightness and spectral class, and a circular stellar disk (or one that is elongated according to the theoretical prediction for its mass and rotation rate). Given the expected and actual transit times it is simple geometry to calculate the impact factor. With additional data (the Rossiter-McLaughlin effect) we can also determine the angle of the projected planetary orbit to the star's polar axis.

Initially it's difficult to accept the transit method as a reliable way to 'see' an exo-planet, until one realizes that the scientific process of elimination (as so elegantly described here) rules out other possibilities. On the other hand, it's quite easy to accept images derived from UV or infrared wavelength.

As I have tried to explain what I'm learning here to *real* n00bies (if you think I ask silly questions sometimes! ), a effective analogy to cite is how rare it is to see a lunar or solar eclipse.

siravan and Mongo are right, but there are a couple of additional complicating factors.

(1) Many planetary orbits are significantly elliptical, so the planet's distance from the star is not very close to the semimajor axis at the time of the transit. Without additional information, this variation is unconstrained, so some transits seem to take up to twice as long as would be possible if the planet were in a circular orbit with the same period.

(2) there are some inconsistencies between the stellar properties as published and my understanding of the relationships between temperature, radius, and mass. This seems to say that there's a lot of variation around the expected means, so it is harder to characterize any particular Kepler candidate and we need, instead, to build a more complex statistical model taking that variability into account.

For example, we have some grazing transits of large planets that imply that a, say, Neptune-sized planet making the same transit would not have adequate SNR for candidacy. That implicitly "blames" the star for the decreased SNR of an off-center transit. It would be preferable to model the star's noise taking into account that each observed candidate has a degree of centrality to its transits ("impact", the Kepler team calls it) which can (for reasons 1 and 2 above) not be determined accurately in any given case.

To handle these factors, it's important to know about typical planetary eccentricity, which radial velocity methods constrain for large planets, but not for small ones, and more about stellar noise, for which the best, newest source of information is Kepler itself.

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